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Philosophy: Faculty Publications Philosophy
6-3-2018
On Conditionals
On Conditionals
Theresa HelkeSmith College, [email protected]
Follow this and additional works at: https://scholarworks.smith.edu/phi_facpubs Part of the Philosophy Commons
Recommended Citation Recommended Citation
Helke, Theresa, "On Conditionals" (2018). Philosophy: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/phi_facpubs/32
ON CONDITIONALS
THERESA SOPHIE CAROLINE HELKE (BA(Hons), Smith College)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHILOSOPHY
NATIONAL UNIVERSITY OF SINGAPORE & YALE-NUS COLLEGE 2018
Supervisors:
Associate Professor Ben Blumson, Main Supervisor
Assistant Professor Malcolm Keating, Co-Supervisor, Yale-NUS College Examiners:
Assistant Professor Robert Beddor Assistant Professor Michael Erlewine
DECLARATION
I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information
which have been used in the thesis.
This thesis has also not been submitted for any degree in any university previously.
__________________________________________ Theresa Sophie Caroline Helke
ACKNOWLEDGEMENTS I’d like to thank
- supervisors Ben Blumson at the National University of Singapore (NUS) and Malcolm Keating at Yale-NUS College;
- original supervisor Jay Garfield, now back at Smith College;
- members of the 7 November 2017 reading group: Bob Beddor, Frank Jackson, Mike Pelczar, Abelard Podgorski and Tang Weng Hong; - Linguistics professor mitcho Erlewine;
- audience members at the February 2018 Israeli Philosophical Association conference in Haifa, December 2017 New Zealand Association of Philosophy conference in Dunedin (especially Max Cresswell and Adriane Rini), November 2017 Australasian Postgraduate Philosophy Conference in Brisbane and March 2016 Kyoto-NUS-Chengchi conference in Kyoto;
- anonymous reviewers at Synthese and the Australasian Journal of Philosophy;
- friends and reviewers Josephine Sedgwick and David Zagoury; and - Cambridge supervisor Steven Methven, now at the Other Place.
Without them, I wouldn’t be submitting this thesis inasmuch as I wouldn’t be submitting one at all, I’d be submitting one on a different topic, or I’d be submitting a version of this thesis with more problems.
TABLE OF CONTENTS
Introduction 1
Chapter 1: The Four Theories 5
Chapter 2: The Election Paradox 34
Chapter 3: The Barbershop Paradox 71
Chapter 4: On Premise Semantics and Apparent Counterexamples to Modus
Ponens and Modus Tollens 105
Conclusion 147
SUMMARY
This thesis is about theories of the indicative conditional and apparent counterexamples to classically valid argument forms. Specifically, it applies the following four theories:
- material (inspired by Grice (1961, 1975 and 1989));
- possible-worlds (inspired by Stalnaker (1981); Lewis (1976); and Kratzer (2012)),
- suppositional (inspired by Adams (1975) and Edgington (1995 and 2014)); and
- hybrid (inspired by Jackson (1987))
to try and solve the following two counterexamples: - Vann McGee’s to modus ponens (1985); and - Lewis Carroll’s to modus tollens (1894).
I argue that none of the theories I consider can explain – without facing any problems – the individually plausible but jointly inconsistent theses that give rise to the apparent counterexamples. A theory can explain a thesis when it can account for why a naïve speaker might have the relevant intuition. In the case of McGee’s Election Paradox, the theses are the following:
- McGee’s argument is invalid;
- McGee’s argument is an instance of modus ponens; and - modus ponens is valid.
Similarly, in the case of Carroll’s Barbershop Paradox, the theses are the following:
- Carroll’s argument is an instance of modus tollens; and - modus tollens is valid.
Despite consensus that the orthodox theories, those I examine, provide answers to questions about conditionals, the paradoxes (even some after over a century) persist: arguments we take obviously to be valid appear to have instances in which they aren’t.
The thesis proceeds in four parts. In Chapter 1, I present the four theories and justify why they rather than others deserve our attention. Along the way, I disagree with Adams (1975). While he claims that on the suppositional theory, we can’t evaluate an indicative conditional whose consequent is a conditional, I include a proof that enables us to.
In Chapter 2, I consider McGee’s counterexample to modus ponens and how the theories might explain the theses of the relevant trilemma. I refute extant solutions: while the Dartmouth group (1986) thinks the material theory can solve the paradox and Edgington thinks the suppositional one can solve it and Jackson thinks the hybrid one can, I show that their solutions fall short of being comprehensive.
In Chapter 3, I do the same, this time considering Carroll’s counterexample to modus tollens. I show that the paradox remains without a solution – and this despite the progress logicians made on conditionals in the centuries following the paradox’s publication and the attempts logicians made at offering a solution.
Finally, in Chapter 4, I examine how premise semantics (Kratzer, 2012), a version of the possible-worlds theory, might explain McGee’s and Carroll’s counterexamples along with some others, these featuring not embedded
conditionals but embedded modals. I argue that, when it comes to counterexamples to modus ponens and modus tollens with (overt) modal verbs and adverbs, the theory gives us some results we want (e.g. invalidating Kolodny and MacFarlane’s Miners examples (2010)) and others we don’t (e.g. validating Cantwell’s Lottery (2008)) – a point which the literature previously overlooked.
LIST OF TABLES
Table 1: Truth conditions of the indicative conditional according to the material theorist
Table 2: Possible combinations for who, between Carter and Anderson, might win if Reagan doesn’t
Table 3: Possible combinations for who, among the barbers, might be in/out Table 4: Breakdown of the blue/red/big/small marbles in the urn
LIST OF FIGURES
Figure 1: Venn diagram of the Election Paradox Figure 2: Venn diagram of the Barbershop Paradox
LIST OF SYMBOLS
É hook (material conditional)
® arrow (indicative conditional)
| given Ù and Ú or ¬ not \ therefore + plus - minus ´ times ÷ divided by = equals
³ greater than or equal to
£ less than or equal to
Ç intersection
È union
Î element of
{p} the singleton set containing p [[p]] the proposition which p expresses
INTRODUCTION
If you’re reading this, then you’re alive. You’re reading this.
Therefore, you’re alive. and
If you’re Theresa Helke, then you wrote this example. You didn’t write this example.
Therefore, you aren’t Theresa Helke.
are instances of modus ponens and modus tollens respectively. The arguments, variations perhaps on the cogito ergo sum, have the following forms:
If A, then B.
A.
Therefore, B.
in the case of the modus ponens and If C, then D.
Not-D.
Therefore, not-C.
in the case of the modus tollens.
The first premise in each is a conditional. Here, in our modus ponens, it’s ‘If A, then B’; and in our modus tollens, ‘If C, then D’. The second premise and conclusion in each do different things. In a modus ponens, the second premise confirms the antecedent of the first, here A, and the conclusion confirms the consequent of the first, here B. In a modus tollens, on the other hand, the second premise contradicts the consequent of the first, here D (i.e.
confirms ‘not-D’) and the conclusion contradicts the antecedent of the first premise, here C (i.e. confirms ‘not-C’).
You might think the two argument forms are valid. When both the premises are true, so too must be the conclusion, i.e. when it’s true both that if you’re reading this, then you’re alive and you’re reading this, then it must be true that you’re alive; likewise, when it’s true both that if you’re Theresa Helke, then you wrote the example but you didn’t write the example, then it must be true that you aren’t Theresa Helke.
Certainly, an introduction to Logic textbook would agree that modus ponens and modus tollens are valid. This thesis is about how they might not be: how arguments can have the forms we saw above and premises we’d accept but a conclusion we’d reject. (Maybe you aren’t alive after all! Maybe you’re me!) Indeed, the research project focuses on theories of the indicative conditional and apparent counterexamples to classically valid argument forms. Specifically, it applies the material (inspired by Grice (1961, 1975 and 1989)), possible-worlds (inspired by Stalnaker (1981); Lewis (1976); and Kratzer (2012)), suppositional (inspired by Adams (1975) and Edgington (1995 and 2014)) and hybrid (inspired by Jackson (1987)) theories to try and solve Vann McGee’s counterexample to modus ponens (1985) and Lewis Carroll’s to modus tollens (1894).
Throughout, I understand an indicative conditional to be like the first – and not the second – of the Adams pair below:
(i) If Oswald didn’t shoot Kennedy, then someone else did.
(ii) If Oswald hadn’t shot Kennedy, then someone else would have (from Khoo (2015, p. 1) who draws on Adams (1970)).
I agree with Justin Khoo (2015) that ‘(i) is about how the world was, given what we now know plus the supposition of the antecedent; (ii) is about how the world would have been (now that we know about it) were its antecedent to have held (pp. 1-2, emphasis in original).
There are both syntactic and semantic differences between (i) and (ii), which I understand to be a subjunctive conditional. The syntactic difference lies in the ‘extra layer of past tense’ (which in English we mark morphologically with the past perfect ‘had’) which (ii) carries and the presence in the consequent of (ii) of the modal auxiliary verb ‘would’ (Khoo, 2015, p. 1).
The semantic difference between (i) and (ii), on the other hand, lies in their truth conditions. Indeed, (i) is true while (ii) is false. We know that someone shot Kennedy and that there was no backup shooter (assuming we’re Warrenites) (Khoo, 2015, p. 1).
If I’m focusing on indicative rather than subjunctive conditionals, it’s because the conditionals in the apparent counterexamples to modus ponens
and modus tollens – the paradoxes – I want to analyse are not subjunctive but indicative.1
My argument is that none of the theories I consider can explain successfully the individually plausible but jointly inconsistent theses that give rise to the apparent counterexamples. Despite consensus that the orthodox theories, those I examine, provide answers to questions about conditionals, the
1 Note that beyond limiting the entire thesis to indicative conditionals, I limit
much of it to indicative conditionals where neither the antecedent nor the consequent contains any (overt) modal. Indeed, it’s only in Chapter 4 that I
paradoxes (even some after over a century) persist: arguments we take obviously to be valid appear to have instances in which they aren’t.
In Chapter 1, I present the four theories and justify why they rather than others deserve our attention. In Chapter 2, I consider McGee’s counterexample to modus ponens (the ‘Election Paradox’) and how the theories might explain the theses of the relevant trilemma. In Chapter 3, I do the same, this time considering Carroll’s counterexample to modus tollens (the ‘Barbershop Paradox’). And in Chapter 4, I examine how premise semantics, (Kratzer, 2012), might explain McGee and Carroll’s counterexamples along with some others.2
2 Note that throughout, I’ll be evaluating the purported counterexamples as
though a single speaker were uttering or considering the propositions. The results might be different if one evaluated each argument as – not a soliloquy but – a dialogue between two speakers. Thanks to mitcho Erlewine for
CHAPTER 1 THE FOUR THEORIES
Consider the following two propositions:
The Equivalence Thesis: The indicative conditional is the material conditional; and
Adams’ Thesis: The assertibility of ‘If Q, then R’ is equal to the conditional probability of R given Q.
For the time being, we understand assertibility of a sentence as the extent to which we are justified in asserting that sentence. We’ll return to this concept in due course. Now, when it comes to accepting or rejecting these two propositions, there are four possibilities:
(i) Accept the Equivalence Thesis but reject Adams’ Thesis; (ii) Reject both the Equivalence Thesis and Adams’ Thesis; (iii) Reject the Equivalence Thesis but accept Adams’ Thesis; and (iv) Accept both the Equivalence Thesis and Adams’ Thesis.
In this chapter, I consider in turn the four possibilities, each corresponding with a theory of the indicative conditional: (i) the material; (ii) the possible-worlds; (iii) the suppositional; and (iv) the hybrid theory. In sections 1 through 4, I’ll outline their main tenets and note some merits and demerits – building a foundation for the chapters to come. In section 5, I conclude.
If these theories deserve our attention here, it’s because they appear to be good candidates for solving the Election and Barbershop paradoxes we’ll see later. The paradoxes challenge classical logic and the theories were built to explain a challenge to classical logic, i.e. that there appears to be some
difference between the material conditional we use to derive proofs and the indicative one we use in everyday speech. Of course, there are other theories: Cantwell (2008), Yalcin (2012), Bledin (2015) to mention only three. If I don’t consider them here it’s not because they don’t deserve our attention. Rather, it’s because considering them lies beyond the scope of this thesis.
1. The material theory
According to the material theorist, the Equivalence Thesis or the ‘horseshoe analysis’ holds: the indicative conditional (which I’ll represent as the arrow ®) is the material conditional (which I’ll represent as the hook É ; an indicative conditional Q ® R is true if and only if the corresponding material conditional Q É R is true. In other words, ‘If Q, then R’ is true if and only if Q
is false or R is true. For example, let Q be ‘Ben’s on sabbatical’ and R be ‘he’s in Australia’. The indicative conditional ‘If Ben’s on sabbatical, then he’s in Australia’ is true if and only if it’s not the case that Ben’s on sabbatical or it’s the case that he’s in Australia.
So, according to the material theorist, we can represent the truth conditions in the following truth table:
Q R Q ® R
True True True
True False False
False True True
Table 1: Truth conditions of the indicative conditional according to the material theorist
Looking at the table, one might think the truth conditions absurd: How can a conditional be true, say, when it has a false antecedent and true consequent? Or when both the antecedent and consequent are false? One way of making sense of the table is through the following story. Imagine one day I promise you ‘If you bring me durian, then I’ll pay you SGD 30’. The following day, when you bring me durian and I pay you SGD 30, I honoured the promise. When I said ‘If you bring me durian, I’ll pay you SGD 30’, I spoke the truth. This accounts for the true-true case. When you don’t bring me any fruit and I don’t pay you any money, I didn’t fail to honour the promise. In asserting the conditional, I spoke the truth. This accounts for the false-false case. Likewise, when you don’t bring me any fruit but (out of magnanimity?) I decide to give you SGD 30, I didn’t fail to honour the promise. In asserting the conditional, I spoke the truth. This accounts for the false-true case. But when you bring me durian and I don’t pay you any money, then I fail to honour the promise. This accounts for the true-false case.
So yes, according to the material theory, an indicative conditional is true if and only if it’s not the case that the antecedent is true and the consequent is false. In other words, an indicative conditional is true if and only if Q is false or R is true.
This point is important in our discussion in Chapters 2 through 4 of
modus ponens and modus tollens. The validity of both argument forms relies on a conditional being false where it has a true antecedent and false consequent.
Next, according to the material theorist, it’s not the case that Adams’ Thesis holds. The material theorist concerns herself with the truth of the antecedent and consequent, rather than the conditional probability of the one given the other. Now, there are two kinds of material theorists. On the one hand, there’s the earlier theorist. According to her, assertability goes with truth: a conditional is assertable if and only if it’s true (Russell, 1905). On the other hand, there’s the later theorist. According to her, assertability doesn’t go with truth.
Logicians developed the later theory in light of objections to the earlier one. The earlier theory faced problems not when it came to conditionals such as the following two:
True-true If 2 is divisible by 2, then 2 is an even number; and
True-false If 2 is divisible by 2, then 2 is an odd number.
Consistent with the theory, we might be prepared to assert the first which, with a true antecedent and consequent, is true. Likewise consistent with the theory, we wouldn’t be prepared to assert the second which, with a true antecedent and false consequent, is false.
No, the earlier theory faced problems when it came to conditionals such as the following two:
False-true If Angela Merkel is the Prime Minister of Singapore, then Angela Merkel is the Chancellor of Germany.
False-false If Vienna is the capital of England, then Vienna is the capital of Switzerland.
Both conditionals are true, according to the earlier (and later) theorist on the grounds that both have a false antecedent. Nonetheless, contra the earlier theorist, we mightn’t be prepared to assert either. We would sound absurd!
Considering these objections, later theorists hold that it’s not the case that the truth of a conditional is necessary and sufficient for our being prepared to assert it. Indeed, we mightn’t be prepared to assert some truth conditionals such as False-true or False-false. Rather, and central to the later theory, we’re prepared to assert an indicative conditional when, doing so, we’d be abiding by the Cooperative Principle.
Before we turn to the principle, note that henceforth when I write ‘material theorist’, I’m referring to the later kind.
1.1. The Cooperative Principle
According to the Cooperative Principle, you must ‘Make your conversational contribution such as is required, at the stage at which it occurs, by the accepted purpose or direction of the talk exchange in which you are engaged’ (Grice, 1975, p. 45).
For example, suppose that you approach me in the street and ask me ‘Where is the wet market?’ To abide by the Cooperative Principle, I could respond: ‘It is at the end of the street on the right’, assuming that the market is indeed at the end of the street on the right. Here, I would be making a contribution such as is required (an answer); I would be doing so at the stage at which it occurs (right after you ask a question); by the accepted purpose or direction of the talk exchange in which we are engaged (addressing your specific question). Not to abide by the Cooperative Principle, I could respond
instead: ‘You could go down the street and turn left’. Here I might be contributing an answer right after you ask a question. However, I wouldn’t be responding to your specific question. Rather, I’d be responding to the question: ‘Where could I go?’
The material theorist is at pains to define the Cooperative Principle because of the existence of conversational implicature. False-true and False-false aside, when speaking, I might make a true statement while implicating a false one. For example, I could say ‘The banana seems yellow’. Assuming that the banana indeed seems yellow, I am speaking the truth. However, by using the verb ‘to seem’ rather than ‘to be’ (‘The banana seems yellow’ rather than ‘The banana is yellow’), I am suggesting that the banana mightn’t actually be yellow. I’m suggesting that the fruit might instead be a different colour.
When it comes to indicative conditionals, assuming that they’re identical to material ones, it’s easy to make true statements while implicating false ones. For example, I could say ‘If the market is not on the left, then it is on the right’. According to the truth table of the material conditional, this statement is true. It has a true consequent. However, truth table aside, I’m implicating a false one. I’m suggesting that the market might be on the left when I know that it is on the right.
Likewise, given a certain scenario, ‘You won’t eat those and live’ (Lewis, 1976, p. 306) is an example of a true but not assertable sentence. Imagine I utter the sentence while pointing at some non-toxic mushrooms and you, deferring to my apparent mycological knowledge, refrain from eating the mushrooms. I told no lie. Formally, the sentence is true. The sentence is a negated conjunction and one of the conjuncts is false (‘you eat those’). Indeed,
it’s not the case that you eat the mushrooms. Nonetheless, the original sentence isn’t assertable. It wouldn’t be a cooperative thing to say.
1.1.1. Maxims
Now, within the Cooperative Principle, there are four categories of maxims by which we abide: Quantity, Quality, Relation and Manner. For the purposes of this dissertation, the following two maxims, one falling under the category of Quality and the other of Manner, are the most relevant:
(i) Don’t assert what you don’t believe (Grice, 1975, p. 46); and (ii) Assert the stronger rather than the weaker (Jackson, 1979, p. 566). Let’s look at each in turn.
(i) Don’t assert what you don’t believe
Grice’s words are ‘Do not say what you believe to be false’ but I think the stronger maxim ‘Don’t assert what you don’t believe’ is desirable. Grice’s maxim allows that, so long as we don’t believe a sentence to be false, we may assert it. This means that according to the maxim, if I don’t believe my mother is in Geneva but I don’t believe my mother’s not in Geneva (maybe she changed her plans and is in Geneva?), I could still assert ‘My mother is in Geneva’. According to the stronger maxim, however, we couldn’t assert ‘My mother is in Geneva’. We’d be asserting something we don’t believe.
(ii) Assert the stronger rather than the weaker
A sentence Q is logically stronger than another R, where Q entails R but not vice versa. For example, let Q be S and R be S Ú ¬S. Here, S entails S Ú ¬ S
but not vice versa. Suppose that S is ‘x is divisible by 2’. From this, it follows that ‘x is divisible by 2 or it is not divisible by 2’. However, from ‘x is divisible by 2 or it is not divisible by 2’, it does not follow that ‘x is divisible by 2’. x could be an odd number. According to (ii), we ought to assert S rather than S Ú ¬S. S is logically stronger than S Ú ¬S.
Of course, there are many possible reasons for wanting to assert R
rather than Q. For example, Q might not be true or believable. ‘Hui Li is teaching’ might not be true while ‘Hui Li is teaching or Hui Li isn’t teaching’ is. Another reason for wanting to assert the weaker R rather than the stronger
Q is that Q might be ‘unduly blunt’ (Jackson, 1979, p. 566). ‘I may or may not fail you’ is less unequivocal than ‘I will fail you’ and we might want to cultivate hope in our student. But abandoning the context of grading and focusing on epistemic and semantic considerations, there is, according to Jackson, no reason not to assert Q: ‘There is no significant loss of probability in asserting [Q] and, by the transitivity of entailment, [Q] must yield everything and more than [R] does. Therefore, [Q] is to be asserted rather than [R], ceteris paribus’ (Jackson, 1979, p. 566).
1.2. Returning to False-true and False-false
The material theorist uses the idea of logically weak and strong statements to explain away counterexamples such as False-true and False-false. When it comes to conditionals A É B where one of ¬ A or B is highly probable, the material theorist says that you should come right out and assert the logically stronger statement, namely ‘¬ A’ or ‘B’ as the case may be (Jackson, 1979, pp. 566-7).
For example, we should assert not False-true but only its consequent. We believe False-true inasmuch as we believe that the consequent is true. The statement ‘Angela Merkel is the Chancellor of Germany’ is logically stronger than ‘If Angela Merkel is the Prime Minister of Singapore, then she is the Chancellor of Germany’. ‘She is the Chancellor of Germany’ entails ‘If Angela Merkel is the Prime Minister of Singapore, then she is the Chancellor of Germany’ but not vice versa. Granted, the conditional is true where the antecedent is false and the consequent true. On the equivalence thesis, however, we should assert only the true consequent.
Likewise, we should assert not False-false but only the negation of its antecedent. We believe False-false inasmuch as we believe that the antecedent is false. The statement ‘It’s not the case that Vienna is the capital of England’ is logically stronger than ‘If Vienna is the capital of England, then Vienna is the capital of Switzerland’. ‘It’s not the case that Vienna is the capital of England’ entails ‘If Vienna is the capital of England, then Vienna is the capital of Switzerland’ but not vice versa. The conditional is true where both the antecedent and consequent are false. On the equivalence thesis, however, we should assert only the negation of the antecedent.
Another way of dismissing False-true and False-false is by saying that we wouldn’t assert them because they are purposefully deceptive or confusing.
False-true suggests that Angela Merkel is the Prime Minister of Singapore.
False-false suggests that Vienna is the capital of England and Switzerland. As we’ll see in the Chapters 2 through 4, this theory faces challenges. Moving forward, we might reject the Equivalence Thesis. Accepting it
required a principle to explain why we mightn’t assert true conditionals. The possible-worlds theorist rejects the Equivalence Thesis. Let’s turn to it next.
2. The possible-worlds theory
Like the material theorist and on the same grounds, the possible-worlds one rejects Adams’ Thesis. She concerns herself with the truth of the antecedent and consequent, rather than the conditional probability of the one given the other. Unlike the material one, however, the possible-worlds theorist furthermore rejects the Equivalence Thesis.
According to the possible-worlds theorist, an indicative conditional ‘If
Q, then R’ is true in a possible world w if and only if R is true in all the Q -worlds which are most similar to w (and vacuously true when there’s no Q -world), where Q-world is one in which the antecedent Q is true (Stalnaker, 1981, pp. 46-7; see also Lewis, 1973, though as the title suggests and unlike us here, Lewis is talking about counterfactual rather than indicative conditionals). For example, the conditional ‘If Nicholas is at the office, then he’s in London’ is true in the actual world just in case Nicholas is in London in all most-similar worlds in which Nicholas is at the office.
And according to the possible-worlds theorist, it’s not the case that we’re prepared to assert an indicative conditional when there’s a high probability in the consequent given the antecedent. Rather, we’re prepared to assert a conditional if and only if it’s true in every antecedent-world most-similar to the actual world.3
3 This isn’t to say the possible-worlds theorist rejects the Equivalence Thesis
Now, possible-worlds theorists disagree on the number of most-similar antecedent-worlds to any given possible world. Some such as Stalnaker (1981) hold that there can be only one (p. 46). Let’s call them the single-world theorists. Others such as Lewis (1973) hold that there can be more than one (pp. 97-8). Let’s call them the multiple-world theorists and look at each in turn, considering their merits and demerits. (As Lewis’s theory is about counterfactual conditionals and here we are looking at indicative ones, the multiple-world theorist is an imagined version of Lewis.)
2.1. Single-world theory
Predictably perhaps, the single-world theorist does a good job accounting for conditionals where there’s a single most-similar antecedent-world. Consider the truth of the conditional A ® B where, say, A is ‘the train arrives late at 5:07pm’ and B is ‘I will miss the connecting 5:03pm train’ and suppose that the train is due to arrive at 5:00pm. This conditional is true. We look at the most-similar A-worlds, those where it’s true that the train arrives at 5:07pm. And in v, the A-world closest to w, the consequent is true. In v, everything is exactly the same as in w with the exception that the train arrives seven minutes late. The connecting train leaves on time at 5:03pm and I miss it. With a true antecedent and a true consequent in the antecedent-world closest to the actual one, the conditional is true.
Of course, there are other worlds in which the train arrives late, at 5:07pm, but I don’t miss the connecting train. For example, in u, the connecting train might leave late, at 5:15pm, allowing me eight minutes to alight from the one train and board the other. In this world, the conditional is
false. It has a true antecedent but a false consequent. However, it doesn’t make the conditional false in the actual world. u is not as close to w as v and we are concerned with the truth value of the conditional in the A-world closest to w. u
is different not only in that the first train arrives seven minutes late but also in that the second train leaves 12 minutes late. Thus we need not worry about the truth value of the conditional in u, only in v.
The single-world theorist does less good a job accounting for conditionals where there’s no most-similar antecedent-world. Such a scenario could arise for a couple of reasons: first, it may be that given any antecedent-world, there’s another, more similar to the actual world; and second, it may be that two or more worlds are equally similar to the actual world and more similar to it than any other worlds (Lewis, 1973, p. 80).
Here are a couple of examples where there’s no most-similar antecedent-world for each of those reasons.
(i) Taller (inspired by Edgington, 1995, p. 252)
If I’m taller than 2.5m, then I must pay to ride the bus.
In the case of the conditional above, for any given antecedent-world, there’s another, more similar one. The more my height tends (from above) toward 2.5m in a possible world, the more similar that world is to the actual world, where I’m shorter than 2.5m. However, there’ll be no most-similar world. For every antecedent-world where I am 2.55m tall, there’s another, more similar world where I’m 2.525m tall. And for every antecedent-world where I’m 2.525m tall, there’s another, even more similar one where I’m 2.5125m tall, etc. Without a most-similar possible world, the theory can’t determine the
truth value of the conditional and without a truth value, the theory can’t determine whether we’d be prepared to assert the conditional.
(Of course, short of being able to determine the truth value, we might think this is a case where the conditional is vacuously true (no Q-world?) – but that isn’t desirable either. As Edgington points out, it would follow that ‘If were taller than I am, no one would know the difference’ would come out as true (1995, p. 252). While this isn’t an indicative conditional, it’s relevant nonetheless as one with potentially no most-similar antecedent world.)
(ii) Les compatriotes and I compatrioti
If François and Italo are compatriots, then they’re both French; and If François and Italo are compatriots, then they’re both Italian.4
In the case of each conditional in the pair above, there are two most-similar antecedent-worlds: the world in which Italo is French and the world in which François is Italian; and in all respects except the nationality of the relevant man, the two worlds are exactly the same as the actual world. With no single most-similar antecedent-world, the single-world theorist can’t determine the truth value of the conditional and, thereby, can’t determine whether we’d be prepared to assert it.
That said, the single-world theorist would take the disjunction of the two (i.e. where Les compatriotes and I compatrioti each form a disjunct) to be true. Indeed and unlike the multiple-world one, the single-world theorist can derive the principle of Conditional Excluded Middle, according to which (Q
® R) Ú (Q ® ¬ R) is a tautology. This will prove valuable in explaining a
thesis (#2) in Chapter 3 (subsection 2.2.). We’ll save her proof of the principle until then.
2.2. The multiple-world theory
In contrast but still predictably perhaps, the multiple-world theorist does a good job accounting for the two cases above. According to her, both conditionals come out as false because for each there are two most-similar antecedent-worlds each assigning a different truth value to the consequent – and the theory considers such conditionals false. That said, the multiple-world theorist doesn’t systematically do better than the single-world one as we’ll see in Chapters 2 and 3. Indeed, her rejection of the principle of Conditional Excluded Middle prevents the multiple-world theorist from explaining a thesis (#2) which the single-world one can in Chapter 3.
Finally and briefly, there’s a third kind of possible-worlds theorist: we’ll call her the premise semanticist. I’ll wait until Chapter 4 to give a full exegesis. The theory offers an account of statements with embedded modals and in that chapter we’ll consider some.
3. The suppositional theory
The suppositional theorist rejects the Equivalence Thesis but accepts Adams’ Thesis. According to her, conditionals are things we evaluate in terms of probabilities (Adams (1975) and Edgington (1995 and 2014)). Indeed, according to her, we’re prepared to assert an indicative conditional Q ® R
supposition that the antecedent Q is true (so long as the probability of Q isn’t equal to zero) – and the Equation holds true.
3.1. The Equation
According to the Equation, the probability of an indicative conditional Q ® R
is equal to the conditional probability of the consequent R, on the supposition that the antecedent Q is true so long as the probability of Q isn’t equal to zero. Formally, writing ‘P’ for probability and ‘|’ for given, we get the following:
P(Q ® R) =P(R | Q) provided that P(Q) ≠ 0.
So, according to the Equation, the probability of the conditional ‘If Josie’s at home, then she’s in New York’ is equal to the conditional probability of ‘Josie’s in New York’ given ‘she’s at home’.
Two things to note at this point:
(i) logicians disagree on the range of Q and R in the Equation. Some take Q
and R to range over only propositions. Adams (1975) and McGee (1989) are cases in point. Others take Q and R to range over propositions and non-propositions (i.e. sentences which don’t necessarily have a truth value). (McGee allows right- but not left-nested conditionals (Hájek, 2012, pp. 150-1) and I’ll side with him.) This disagreement is an important matter, as we’ll see when deriving the Triviality Result below and when defining modus ponens
and modus tollens in Chapters 2 and 3; and
(ii) the Equation is different from Adams’ Thesis: while the Equation equates probability of a conditional with conditional probability, Adams’ Thesis equates assertibility with conditional probability. In Chapter 2 (subsection 4.1), we’ll see the difference prevent the hybrid theorist from adopting as it is
a proof by the suppositional theorist. The proof relies on the Equation which, unlike the suppositional one, the hybrid theorist doesn’t accept. Indeed, the difference will require the hybrid theorist to adapt the proof.
Now, to calculate the conditional probability of the consequent, given the antecedent, we rely on the Ratio Formula.
3.2. The Ratio Formula
According to the Ratio Formula, for any proposition Q and R, the conditional probability of R given Q equals the probability of the conjunction of Q and R
divided by the probability of Q so long as the probability of Q isn’t equal to zero(Hájek 273). Formally, we get
P(R | Q) =P(Q Ù R) / P(Q) provided that P(Q) ≠ 0.
So, the conditional probability of ‘Josie’s in New York’ given ‘she’s at home’ is equal to the probability of ‘Josie’s at home and she’s in New York’ divided by the probability of ‘Josie’s in New York’.
3.3. Returning to False-true & co and evaluating the theory
The suppositional theorist can account for the conditionals we’ve been considering. According to her, we aren’t prepared to assert False-true because, assuming that Angela Merkel is the Prime Minister of Singapore, there won’t be a high probability that she’s the Chancellor of Germany. Among the facts of the world is the following one: Singapore and Germany are two different countries and no person is simultaneously the head of state in both.
We aren’t prepared to assert False-false because assuming that Vienna is the capital of England, there won’t be a high probability that Vienna is the
capital of Switzerland. Also among the facts of the world is the following one: England and Switzerland are two countries and no city is simultaneously the capital of both.
The suppositional theory can account for Taller the same way in which it accounts for other conditionals. We’d evaluate the consequent of Taller on the supposition of the antecedent.
Finally, consistent with our intuition, we wouldn’t be prepared to assert Les compatriotes or I compatrioti. Assuming François and Italo are compatriots, there wouldn’t be a high probability that they were both French, or in the other case, that they were both Italian.
The suppositional theorist does a good job explaining why we would or wouldn’t assert or could evaluate False-true and the other conditionals we’ve considered so far. However, the theory makes a serious departure from the notions of truth and validity of classical logic – which some might view as a demerit – and without that departure, the Triviality Result follows (Lewis, 1976).
3.4. The Triviality Result
The Equation in part implies that a conditional is non-propositional. Indeed, the Equation plus the assumption that a conditional is propositional would lead to absurdity: they imply that no change in any proposition’s probability affects another’s. For example, an increase in the probability that Roger Federer will win the Wimbledon final against Rafael Nadal (say, Roger’s serving match point, leading six games to four in the third set and two sets to zero) doesn’t
affect the original probability that Rafael will win. Logicians call such a scenario, where no change affects any change, trivial.
Specifically, the scenario arises from the fact that conditionals could embed everywhere and the probability of any conditional would be the probability of its consequent. For any sentence ‘Roger will win’, we could create a conditional ‘If x, then Roger will win’ and no matter what we choose for x, the probability of the conditional will be that of the consequent ‘Roger will win’. x could be ‘Roger will lose’ and that wouldn’t affect the probability of the conditional. Absurdly, it would remain that of ‘Roger will win’.
Now, to derive the Triviality Result, apart from relying on the Ratio Formula, the Equation and Conditionalisation (which we’ll define), we rely on the following axioms and theorems:
Axioms
A1. P(A) ³ 0
A2. If T is a tautology, then P(T) = 1
A3. P(A Ú B) =P(A) +P(B) if A and B are inconsistent
Theorems
T1. If C is a contradiction, then P(C) = 0 T2. P(A) =P(B) if A and B are equivalent
T3. P(A Ù B) =P(A | B) ´ P(B) provided that P(B) ≠ 0
We begin by proving a lemma – which will serve us not only in this proof but later.
3.4.1. Import-Export
We call Import-Export the rule of inference according to which we can derive
Q ® (R ® S) from (Q Ù R) ® S and vice versa. We’ll also call Import-Export the probabilistic equivalence between the probabilities of those two sentences:
P(Q ® (R ® S) =P((Q Ù R) ® S). Before we derive this equivalence, we note that Import-Export will prove important in this thesis.
In Chapter 2, for example, the possible-worlds theorist will appeal to her rejection of Import-Export to reject in turn the proposition according to which McGee’s argument is an instance of modus ponens. While McGee formalises the first premise as having the form Q ® (R ® S), the possible-worlds theorist would formalise it as having the form (Q Ù R) ® S and, citing the invalidity of Import-Export according to her theory (i.e. rejecting that one can derive the one relevant conditional from the other), she would deny that McGee’s argument is counterexample to modus ponens.
Also in that chapter, as in Chapter 3, the suppositional theorist will appeal to her acceptance of Import-Export to calculate the probability of the first premise in McGee and Carroll’s respective arguments. Contra Adams (1975, pp. 30-3), I show that she can evaluate the probability of a conditional with a conditional as consequent: by evaluating the probability of the equivalent conditional with a conjunction as antecedent.
McGee notes that his argument suggests it’s not the case modus ponens
and Import-Export can both be valid where the conditional connective is stronger than É, as we’d expect ® to be (McGee, 1985, pp. 465-6). While we find both valid where the connective is the material conditional, ‘[w]e have
explicit examples to show that the indicative conditional does not satisfy modus ponens’, McGee writes (1985, p. 466). I’ll dedicate Chapter 2 to examining one of the examples he offers.
Here, first, to prove the probabilistic equivalence of Q ® (R ® S) and (Q Ù R) ® S, we rely on not only the Equation but also Conditionalisation.
3.4.1.1. Conditionalisation
According to Conditionalisation, for any formulae Q and R, the probability of
R after we find out that Q is equal to the conditional probability of R given Q. Formally, we get the following equation, where PQ(R) is the posterior probability of R after we discover Q (Lewis, 1976, p. p. 299):
PQ(R) =P(R | Q)
So, according to Conditionalisation, the probability of ‘Josie’s in New York’ after we find out that ‘she’s at home’ is the conditional probability of ‘she’s in New York’ given ‘Josie’s at home’.
Now, assuming P(Q Ù R) ≠ 0 and using the Equation and Conditionalisation, we can prove that Q ® (R ® S) and (Q Ù R) ® S are probabilistically equivalent (see Alan Hájek’s proof in Bennett, 2003, p. 62):
1. P(Q ® (R ® S)) =P((R ® S) | Q) by the Equation
2. =PQ(R ® S) from 1 by Conditionalisation 3. =PQ(S | R) from 2 by the Equation 4. =PQ(S Ù R) ÷PQ(R) from 3 by the Ratio Formula 5. =P((S Ù R) | Q) ÷P(R | Q) from 4 by Conditionalisation 6. = [P(S Ù R Ù Q) ÷P(Q)] ÷ [P(R Ù Q) ÷ P(Q)]
7. =P(S Ù R Ù Q) ÷P(R Ù Q) from 6 by cancellation of P(Q) 8. =P(S | (Q Ù R)) from 7 by the Ratio Formula 9. =P((Q Ù R) ® S) from 8 by the Equation Taking it from top to bottom and still assuming P(Q Ù R) ≠ 0,
10. P(Q ® (R ® S)) =P((Q Ù R) ® S)
from 1 and 9 by transitivity of identiy
which is Import-Export.
The suppositional theorist welcomes this result. The equivalence allows her to evaluate the probability of a conditional whose consequent is itself a conditional. Without the result, she couldn’t as she can evaluate only a conditional whose antecedent and consequent are propositional, and a conditional isn’t propositional.
Here, assuming the suppositional theorist can derive the proof, I disagree with Adams (1975). He claims that on the suppositional theory we can’t evaluate conditionals whose consequents are conditionals (Adams, 1975, pp. 30-3). This proof gives us a means to do so.
Next, we turn to prove that for any conditional Q ® R, the probability of that conditional is equivalent to the probability of the consequent: the Triviality Result proper.
3.4.2. Triviality
Proof (from Bennett, 2003, p. 63. Bennet himself takes the proof from Blackburn, 1986, pp. 218-20):
12. =P((R Ù (Q ® R)) Ú (¬R Ù (Q ® R))) from 11 by T2 13. =P(R Ù (Q ® R)) +P(¬R Ù (Q ® R)) from 12 by A3 14. = [P((Q ® R) | R) ´ P(R)] + [P((Q ® R) | ¬R) ´ P(¬R)] from 13 by T3 15. = [P(R ® (Q ® R)) ´ P(R)] + [P(¬R ® (Q ® R)) ´ P(¬R)] from 14 by the Equation 16. = [P((R Ù Q) ® R) ´ P(R)] + [P((¬R Ù Q) ® R) ´ P(¬R)]
from 15 by Import-Export 17. = [P(R | (R Ù Q)) ´ P(R)] + [P(R | (¬R Ù Q)) ´ P(¬R)]
from 16 by the Equation
18. = [(P(R Ù R Ù Q) ÷P(R Ù Q)) ´ P(R)] + [(P(R Ù ¬R Ù Q) ÷P(¬R
Ù Q)) ´ P(¬R)]
from 17 by the Ratio Formula 19. = [1 ´ P(R)] + [0 ´ P(¬R)] from 18 by algebra, T1 and T2 20. =P(R) from 19 by algebra
3.4.3. Avoiding the result
Now, to avoid the Triviality Result while still holding onto the Equation, the suppositional theorist has various options, some more attractive than others. One option is to reject 1. She might say the Equation doesn’t hold where Q
and R ranger over non-propositions and here there’s nothing preventing Q or R
Another option is to reject the move from 11 to 12. This step seems to assume that a conditional is propositional and the suppositional theorist rejects that assumption. The step assumes that the probability theorem T2 applies but, for the theorem to apply, the conditionals would have to be logically equivalent and in 11 they’re merely probabilistically so, meaning the theorem doesn’t apply.
A third option to avoid the Triviality Result is to reject the move from 12 to 13. This step also seems to assume that a conditional is propositional. The step assumes that the probability axiom A3 applies but, for the axiom to apply, the conditionals would have to be logically inconsistent and here they’re merely probabilistically so, meaning the axiom doesn’t apply. To be logically inconsistent, the conditionals would have to have truth values and, according to the suppositional theorist, they don’t.
Here, the second and third options are more attractive than the first. While rejecting 1 prevents the suppositional theorist from deriving the proof of Import-Export, rejecting the move from 11 to 12 or 12 to 13 doesn’t – and Import-Export will serve the suppositional theorist later, as we’ll see in Chapters 2 and 3, when she tries to explain our intuitions in the case of apparent counterexamples to modus ponens and modus tollens.
Either way, according to the suppositional theorist, two propositions don’t combine into a single proposition we judge as probably true when we judge the second to be probably true on the supposition of the first (Edgington, 1995, p. 305). So, according to the suppositional theory it’s not the case that ‘If Josie’s at home, then she’s in New York’ can be true or false.
Likewise, according to the suppositional theorist, it’s not the case that classical validity is relevant when we’re talking about arguments with indicative conditionals. Classical validity concerns itself with truth-preservation and here we’re taking indicative conditionals to be non-propositional. Rather, when we’re talking about arguments with indicative conditionals, probabilistic validity is relevant.
To define probabilistic validity, we define uncertainty. We define the uncertainty of a formula Q as one minus the probability of Q (Adams, 1975, p. 3). Formally, writing ‘U’ for uncertainty, we get
U(Q) = 1 - P(Q)
Using this definition of uncertainty, we define an argument as probabilistically valid when the uncertainty of the conclusion can’t exceed the sum of the uncertainties of the premises (Bennett, 2003, p. 129; Adams, 1975, pp. 1-2). In contrast to classical validity which bars falsity from entering ‘along the way from the [true premises] to the conclusion’, probabilistic validity bars improbability from entering along the way (Bennett, 2003, p. 129).
The departure from notions of truth and validity we know in classical logic will prove helpful for the theory when explaining the intuitions we might have when it comes to the Election and Barbershop paradoxes, as we’ll see in Chapters 2 and 3. Still, some might view the departure as a demerit for the theory, given the consequences we’ve seen above.
4. The hybrid theory
According to the hybrid theorist, the Equivalence Thesis and Adams’ Thesis both hold true. The hybrid theory combines elements from the material theory
and the suppositional theory. On the one hand, like the material theory, the hybrid one concerns itself with truth. On the other hand, like the suppositional one, the hybrid theory concerns itself with probability. For example, an indicative conditional is true where the corresponding material one is; and it’s assertible where the conditional probability of the consequent given the antecedent is high. The hybrid theorist speaks of an assertible conditional as being robust: it’s such that we wouldn’t abandon belief in it upon learning that its antecedent is true (Jackson, 1979, pp. 569-70).
4.1. Assertibility
Note that the material theory’s concept of assertability with an ‘a’ and the hybrid theory’s concept of assertibility with an ‘i’ aren’t the same. The difference in vowels reflects a difference in meaning. The assert-a-bility of a sentence depends on ‘local’ factors: ‘how important and relevant is the information to present concerns, is the information already widely known, and so on and so forth?’ (Jackson, 1987, p. 11). In contrast, the assert-i-bility of a sentence depends on factors ‘governing when it is justified or warranted – in the epistemological sense, not in a purely pragmatic one – to assert it, or, as this comes in degrees, to what extent it is justified to assert it under the circumstances’ (Jackson, 1987, p. 8).5
While assertibility and probability of truth go hand in hand for many sentences, they don’t for many others, notably indicative conditionals. Let’s
5 For those familiar with the Gricean contrast between conversational
implicature and conventional implicature, I’ll add that assertability is related to conversational implicature while assertibility is related to conventional implicature. So here in the hybrid theory subsection, I’m talking about
take a simpler example – one whose main connective is not a conditional but a conjunction – first. Consider the following sentence:
Ming Bin is a good student even though he’s intelligent.
The sentence has a low assertibility but a high probability: it has a high probability if and only if the conjunction ‘Ming Bin is a good student and is intelligent’ has a high probability – and arguably, it does. The sentence has a low assertibility, however, given the presence of ‘even though’. The phrase places an ‘additional [assertibility] requirement’ on the sentence, over and above the probability requirement – and the additional requirement isn’t satisfied here (Jackson, 1987, p. 60).
As in the example, in the case of indicative conditionals, assertibility doesn’t go hand in hand with probability. The assertibility of an indicative conditional depends on the conditional probability of its consequent given its antecedent (Jackson, 1987, p. 11). Formally, we get Adams’ Thesis, which we saw at the very beginning and according to which the assertibility of Q ® R is equal to the conditional probability of R given Q so long as the probability of
Q isn’t equal to zero.
If this weren’t the case (i.e. if the assertibility of indicative conditionals
did go hand in hand with probability), then the assertibility of ‘If it’s not the case that it’s raining, then it’s raining’ would be whatever the probability of rain were, since the probability of the conditional would be the probability of the disjunction ‘It’s raining or it’s raining’ which in turn would be the probability of the disjunct ‘it’s raining’.
As it is, the assertibility of the conditional ‘If it’s not the case that it’s raining, then it’s raining’ is zero. The conditional probability of ‘it’s raining’
given ‘it’s not the case that it’s raining’ is zero, since according to the Ratio Formula, the probability of the consequent given the antecedent is equal to the probability of ‘it’s not the case that it’s raining and it’s raining’ divided by ‘it’s raining’. Here, since the numerator is the probability of a contradiction, it has a probability of zero and, regardless of the denominator, the quotient (i.e. the conditional probability of ‘it’s raining’ given ‘it’s not the case that it’s raining’) will be zero.
Note further that, in certain contexts, a conditional might be assert-i-ble but not assert-a-assert-i-ble. In a silent reading room, ‘If my birth certificate is correct, Jill is my mother’ is a case in point. Granted, generally, it might make sense for me to say it because there’s a high probability that Jill is my mother, given that my birth certificate is correct. However, it wouldn’t make sense for me to say it in a room where I’m not permitted to speak (Jackson, 1987, pp. 10-1).
4.2. Returning to False-true & co and evaluating the theory
According to the hybrid theory, we can dismiss both False-true and False-false on the grounds that they aren’t robust. We would abandon belief in
False-true upon learning that its antecedent is true. Supposing it true that Angela Merkel is the Prime Minister of Singapore, we would not believe that she is also the Chancellor of Germany. Singapore and Germany are two different countries and one person is not simultaneously the head of state in both. Likewise, we’d abandon belief in False-false upon learning that its antecedent is true. Supposing it true that Vienna is the capital of England, we would not believe that Vienna is also the capital of Switzerland. England and
Switzerland are two different countries and one city is not simultaneously the capital of both.
As to Taller, Les compatriotes and I compatrioti, the hybrid theorist would evaluate them in the same way as a suppositional one.
Among the merits of the theory are the following two: first, of course, it can explain the sentences; and second, it aligns with the truth-conditional theory of meaning. Indeed, it doesn’t face the problem the suppositional theory did when it comes to aligning with the truth-conditional theory of meaning. According to the hybrid theory, an indicative conditional can be true – and so, on the truth conditional-theory of meaning, it is possible for us to know its meaning. We need only know the conditions under which the conditional is true.
As to the demerits, there’s the following one: it requires the seemingly ad hoc adoption of rules to account for Import-Export preserving assertibility (i.e. the idea that the assertibility of Q ® (R ® S) is equivalent to the assertibility of (Q Ù R) ® S). I discuss this as the matter arises in Chapters 2 and 3.
5. Concluding
In this chapter, I outlined the topic of the thesis: theories of the indicative conditional and apparent counterexamples to classically valid argument forms; and I considered in turn four possible positions when it comes to accepting or rejecting the Equivalence Thesis or Adams’ Thesis, each position corresponding with a theory of the indicative conditional: the material; the possible-worlds; the suppositional; and the hybrid theory.
Along the way, I disagreed with Adams (1975). While he claims that on the suppositional theory, we can’t evaluate an indicative conditional whose consequent is a conditional, I included a proof that enables us to.
CHAPTER 2
THE ELECTION PARADOX
So far, we’ve seen various theories of the indicative conditional coming under four headings: the material, possible-worlds, suppositional and hybrid theories. Next, we turn to apply those theories to an alleged counterexample to an apparently valid argument form. In this chapter, I proceed largely from first principles in analysing Vann McGee’s alleged counterexample to modus ponens (1985). I focus on responses the theories would offer to the trilemma McGee’s argument presents.
The counterexample takes as context the 1980 US presidential election. Shortly before voting day, opinion polls reveal five things:
(i) the Republican Ronald Reagan is in first place; (ii) the Democrat Jimmy Carter is in second; (iii) another Republican John Anderson is in third; (iv) Reagan is substantially ahead of Carter; and (v) Carter is substantially ahead of Anderson. Intuitively, the following two premises seem true:
If a Republican wins the election, then if it’s not Reagan who wins it will be Anderson; and
A Republican will win the election.
The first seems true inasmuch as there are two Republicans in the race; and, logically, if a Republican wins and the one Republican loses, the other one will win. Moreover, the second premise seems true inasmuch as the polls
show the Republican Reagan in first place and far ahead of Democrat Carter in second.
Also intuitively, however, the conclusion seems false even though it follows by modus ponens from the two premises:
If it’s not Reagan who wins, it will be Anderson (McGee, 1985, p. 462).
The conclusion seems false inasmuch as the polls show Carter in second position – not Anderson. If Reagan loses, then presumably Carter, in second position, would win.
This appears to be a counterexample to modus ponens. Writing A for ‘a Republican wins’, B for ‘Reagan doesn’t win’, C for ‘Anderson wins’, ‘®’ for the indicative conditional connective and ‘\’ for ‘therefore’, we get the following argument6:
A ® (B ® C)
A
\ B ® C
The premises seem true and the second premise A is the antecedent of the first premise A ® (B ® C). The conclusion B ® C, however, seems false even though it’s the consequent of the first premise. Indeed, there are two possible combinations if Reagan doesn’t win and between those two, it’s not necessarily the case that it will be Anderson:
6 Here, I’m departing from McGee’s original wording. I’m saying e.g. the
consequent of the conclusion is ‘Anderson wins’ rather than ‘it will be Anderson’. I don’t depart from the wording everywhere henceforth and, even if I did, I don’t think the departure has any significant implication except,
Reagan Carter Anderson
1 Loses Wins Loses
2 Loses Loses Wins
Table 2: Possible combinations for who, between Carter and Anderson, might win if Reagan doesn’t
It’s possible – and, given the results of the poll, likely – that if it’s not Reagan who wins, it will be Carter.
I’m not alone in offering a discussion of McGee. Walter Sinnott-Armstrong, James Moor and Robert Fogelin (1986) (later ‘the Dartmouth group’), E.J. Lowe (1987), Christian Piller (1996), Bernard D. Katz (1999), Joseph S. Fulda (2010) and most recently Justin Bledin (2015) do so too. I’ll cite or discuss some where relevant. That said, I won’t discuss Bledin’s positive view, for example, as his informational theory of logic doesn’t fit within the theories of conditionals I’m considering.
In this chapter, I refute extant solutions: while the Dartmouth group thinks the material theory can solve the paradox and Edgington thinks the suppositional one can solve it and Jackson thinks the hybrid one can, I’ll show that their solutions fall short of being comprehensive.
I proceed by considering McGee’s argument in light of four theories of the indicative conditional: the material (inspired by Grice 1961, 1975 and 1989), possible-worlds (inspired by Stalnaker (1981) and Lewis (1976)), suppositional (inspired by Adams (1975) and Edgington (1995 and 2014)) and hybrid (inspired by Jackson (1987)). Specifically, I examine how the theories would solve an inconsistent triad. The three individually plausible but jointly inconsistent theses are the following:
#1 the premises are true and the conclusion is false;
#2 the argument is an instance of modus ponens, i.e. is one with the form Q ® R; Q; \ R, where one has replaced the letters with propositions; and
#3 modus ponens is valid, i.e. in any argument with the form Q ® R;
Q; \ R, the truth of the premises guarantees the truth of the conclusion.
As we’ll see, not all theories can explain the plausibility of all three theses at once – and to solve the trilemma they must. Moreover, the theories which can explain all three theses face problems of their own.
In section 1, I present the material theory and show that it rejects #1 but can’t explain the plausibility of the thesis without undermining itself. In section 2, I show that the possible-worlds theory rejects #1 too but can’t explain its plausibility. In section 3, I present the suppositional theory and show that it rejects all three theses but can explain at most the plausibility of two. In section 4, I present the hybrid theory and show that while it rejects #1, it can explain the plausibility of the thesis and while the theory accepts #2 and #3, it fails in its original mission. In section 5, I conclude.
1. The material theory
Recall that according to the material theorist, an indicative conditional Q ® R
is true if and only if the corresponding material conditional Q É R is true. In other words, ‘If Q, then R’ is true if and only if Q is false or R is true. For example, let Q be ‘Ben’s on sabbatical’ and R be ‘he’s in Australia’. The indicative conditional ‘If Ben’s on sabbatical, then he’s in Australia’ is true if
and only if it’s not the case that Ben’s on sabbatical or it’s the case that he’s in Australia.
In this section, I show how the material theory accepts Theses #2 as well as #3 but rejects #1 and can explain its plausibility. I show also that in explaining the plausibility of #1, the material theory deals itself a blow: it finds unassertable a sentence we might assert.
1.1. Accepting #2 and #3 but rejecting #1
According to the material theorist, the following holds true:
- the argument is indeed an instance of modus ponens. We can reduce it to the form Q ® R; Q; \ R;
- modus ponens is valid. It’s impossible for the premises to be true and the conclusion false; and
- both premises are true – and so is the conclusion. Given the results of the poll, the second premise is true. So is the conclusion inasmuch as the antecedent of the conditional is false. And the first premise is true inasmuch as the consequent of the conditional is true: it’s the conclusion and, as we’ve just seen, the conclusion is true.
The material theorist can explain nonetheless the plausibility of Thesis #1 by appealing to the concept of assertability. Recall from Chapter 1 that according to the Cooperative Principle, some sentences are true but not assertable. Given a certain scenario, ‘You won’t eat those and live’ is a case in point (Lewis, 1976, p. 306). Imagine I say the sentence while pointing at some non-toxic mushrooms and you, deferring to my apparent mycological
